Convergence of the mac scheme for variable density flows

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Convergence of the mac scheme for variable density flows

Thierry Gallouët, Raphaele Herbin, Jean-Claude Latché, K Mallem

To cite this version:

Thierry Gallouët, Raphaele Herbin, Jean-Claude Latché, K Mallem. Convergence of the mac scheme

for variable density flows. Finite Volume for Complex Applications 8, Jun 2017, Lille, France. �hal-

01483467�

CONVERGENCE OF THE MAC SCHEME FOR VARIABLE DENSITY FLOWS

T. GALLOU¨ET, R. HERBIN, J.C. LATCH´E, AND K. MALLEM

Abstract. We prove in this paper the convergence of an semi-implicit MAC scheme for the time-dependent variable density Navier-Stokes equations.

1. Introduction

Let Ω be a parallelepiped ofR^d, withd∈ {2,3}and T >0, and consider the following variable density Navier-Stokes equations posed on Ω×(0, T):

∂tρ¯+ div(¯ρu) = 0,¯ (1a)

∂t(¯ρu) + div(¯¯ ρu¯⊗u)¯ −∆¯u+∇p¯=f, (1b)

div ¯u= 0, (1c)

where ¯ρ, ¯uand ¯pare the density, the velocity and the pressure of the flow andf ∈L²(0, T;L²(Ω)^d). This system is complemented with initial and boundary conditions ¯u|∂Ω= 0,u¯|t=0 =u0, ¯ρ|t=0 =ρ0, which are such thatρ0∈L^∞(Ω), 0< ρmin< ρ0≤ρmax andu0∈L²(Ω)^d. A pair (¯ρ,u) is a weak solution of problem¯ (1) if it satisfies the following properties:

– ρ¯∈ {ρ∈L^∞(Ω×(0, T)), ρ >0 a.e.in Ω×(0, T)}.

– u¯ ∈ {u∈L^∞(0, T;L²(Ω)^d)∩L²(0, T;H₀¹(Ω)^d), div u= 0a.e.in Ω×(0, T)}. – For allϕinC_c^∞(Ω×[0, T)),

(2) −

Z T 0

Z

Ω

¯

ρ∂tϕ+ ¯ρu¯· ∇ϕdxdt= Z

Ω

ρ0(x)ϕ(x,0) dx.

– For allv in{w∈C_c^∞(Ω×[0, T))^d,divw= 0}, (3)

Z T 0

Z

Ω

−ρ¯u¯·∂tv−(¯ρu¯⊗u) :¯ ∇v+∇u¯ :∇v

dxdt= Z

Ω

ρ0u0·v(·,0) dx+ Z T

0

Z

Ω

f ·v dxdt.

The existence of such a weak solution was proven in [9]; convergence results exist for the discontinuous Galerkin approximation [8] and for a finite volume/finite element scheme [7]. Here we prove the convergence of the MAC scheme.

2. The numerical scheme

LetM be a MAC mesh (see e.g. [4] and Figure 1 for the notations). The discrete pressure and density unknowns are associated with the cells of the mesh M, and are denoted by

ρK, K ∈M and

pK, K ∈ M . The discrete velocity unknowns approximate the normal velocity to the mesh faces, and are denoted (uσ)_σ∈E⁽ⁱ⁾,i∈

|1, d|

, whereEis the set of the faces of the mesh, andE⁽ⁱ⁾ the subset of the faces orthogonal to the i-th vector of the canonical basis of R^d. We define E_ext = {σ ∈ E, σ ⊂ ∂Ω}, E_int = E\E_ext, E⁽ⁱ⁾

int=E_int∩E⁽ⁱ⁾andE⁽ⁱ⁾

ext =E_ext∩E⁽ⁱ⁾. The regularity of the mesh is defined by:

η^M= max|σ|

|σ^′|, σ∈E⁽ⁱ⁾, σ^′ ∈E^(j), i, j∈

|1, d|

, i6=j ,

and we denote byh^M the space step. The discrete spaceL^M for the scalar unknowns (i.e. the pressure and the density) is defined as the set of piecewise constant functions over each of the grid cellsK ofM, and the

2000Mathematics Subject Classification.

1

Dσ

K

L

σ=K|L σ^′′

×

×

× x_σ^′

xσ xσ^′′

ǫ2 ǫ3

σ^′ ǫ1=σ|σ^′

∂Ω

dǫ3

Dǫ3

dǫ2

dǫ1

Figure 1. Notations for control volumes and dual cells.

discrete space for thei^th velocity component, HE(i), as the set of piecewise constant functions over each of the grid cells Dσ, σ ∈ E⁽ⁱ⁾. The set of functions of L^M with zero mean value is denoted byL^M,0. As in the continuous case, the Dirichlet boundary conditions are (partly) incorporated into the definition of the velocity spaces:

HE(i),0=n

u∈HE(i), u(x) = 0 ∀x∈Dσ, σ∈E⁽ⁱ⁾

ext

o, for 1≤i≤d (i.e. we imposeuσ = 0 for allσ∈E_ext). We then setH^E,0=Qd

i=1HE⁽ⁱ⁾,0.

Let 0 = t0 < t1 < · · · < tN =T be a partition of the time interval (0, T), with δt = tn+1−tn. Let {uⁿ⁺¹_σ , σ∈E⁽ⁱ⁾,0≤n≤N−1, 1≤i≤d},{pⁿ⁺¹_K , K ∈M, 0≤n≤N−1}and{ρⁿ⁺¹_K , K ∈M,0≤n≤N−1} be the sets of discrete velocity, pressure and density unknowns. Defining the characteristic function1A of any subsetA⊂Ω by 1A(x) = 1 if x∈A and 1A(x) = 0 otherwise, the corresponding piecewise constant functions for the velocities are of the form:

ui=

N−1X

n=0

X

σ∈E⁽ⁱ⁾

int

uⁿ⁺¹_σ 1Dσ1]t_n,tn+1],

andXi,E,δtdenotes the set of such piecewise constant functions on time intervals and dual cells; we then set XE,δt=Qd

i=1Xi,E,δt. The pressure and density discrete functions are defined by:

p=

NX−1

n=0

X

K∈M

pⁿ⁺¹_K 1K1]tn,tn+1], ρ=

N−1X

n=0

X

K∈M

ρⁿ⁺¹_K 1K1]tn,tn+1],

andY^M,δt denotes the space of such piecewise constant functions. The numerical scheme reads:

Initialization: u⁽⁰⁾=ePEu0, ρ⁽⁰⁾ =PMρ0. (4a)

For 0≤n≤N−1, solve for uⁿ⁺¹∈H^E,0, ρⁿ⁺¹∈L^M andpⁿ⁺¹∈L^M,0: ð_tρⁿ⁺¹+ div^M(ρⁿ⁺¹uⁿ) = 0,

(4b)

ð_t(ρu)ⁿ⁺¹+CE(ρⁿ⁺¹uⁿ)uⁿ⁺¹−∆^Euⁿ⁺¹+∇^E pⁿ⁺¹=fⁿ⁺¹E , (4c)

div^Muⁿ⁺¹= 0, (4d)

with the interpolators and discrete operators defined as follows.

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Grid interpolators– The Fortin interpolator is defined by Pe_Eu = (Pe_E(i))i=1,...,d with Pe_E(i) : H₀¹(Ω) −→

HE(i),0and

vi7−→Pe_E(i)vi= X

σ∈E(i)

vσ 1Dσ withvσ= 1

|σ| Z

σ

vi dγ(x), σ∈E⁽ⁱ⁾.

Forq∈L²(Ω), PMq∈L^M is defined byPMq(x) = 1

|K| Z

K

q dxforx∈K.

Discrete time derivative – Forρ∈Y^M,δt, ð_tρ∈Y^M,δt is defined by:

ð_tρ(x, t) =

N−1X

n=0

ð_tρⁿ⁺¹(x)1]tn,tn+1](t) with ð_tρⁿ⁺¹= X

K∈M

1

δt (ρⁿ⁺¹_K −ρⁿ_K)1K.

Discrete divergence– Let uK,σ be defined as uK,σ=uσnK,σ·ei for any faceσ∈E⁽ⁱ⁾,i= 1, . . . , d. The discrete (upwind finite volume) divergence operator div^M is defined by:

div^M: L^M×H^E,0−→L^M, (ρ,u)7→div^M(ρu) = X

K∈M

1

|K| X

σ∈E(K)

FK,σ 1K,

withFK,σ=|σ|ρσuK,σ forK∈M, σ=K|L∈E(K), andρσ =ρK ifuK,σ≥0,ρσ=ρL otherwise. For all K∈M, we set (divu)K = div(1×u)K.

Pressure gradient operator– The discrete pressure gradient operator is defined as the transpose of the divergence operator, so∇^E: L^M−→H^E,0, p7→ ∇^E(p) with:

(5) ∇^Ep= X

σ=K|L∈E⁽ⁱ⁾

int

(ðp)σnK,σ 1Dσ, with (ðp)σ= |σ|

|Dσ|(pL−pK).

Discrete Laplace operator – The discrete diffusion operator ∆^E is defined in [4] and is coercive in the sense that−R

Ω∆^Ev·v dx=kvk²1,E,0for anyv∈H^E_m,0, wherek · k1,E,0 is the usual discrete H¹-norm ofu (see [4]). This inner product may also be formulated as theL²-inner product of adequately chosen discrete gradients [4].

Discrete convection operator – The numerical convection fluxes and the approximations of ρ in the momentum equation are chosen so as ensure that a discrete mass balance holds on the dual cells, in order to recover a discrete kinetic energy inequality. This idea was first introduced in [3, 1] for the Crouzeix-Raviart and Rannacher-Turek scheme, in [6] for the MAC scheme and was adapted to a DDFV scheme [5]. For ǫ=σ|σ^′, the convection fluxR

ǫρuiu·nσ,ǫ dγ(x) is approximated byFσ,ǫuǫ, where uǫ = (uσ+uσ^′)/2 and Fσ,ǫ is the numerical mass flux throughǫ outwardDσ defined as follows:

- First case – The vectorei is normal to ǫ, and ǫ is included in a primal cellK. Then the mass flux throughǫ=σ|σ^′ is given by:

Fσ,ǫ= 1

2 FK,σ nDσ,ǫ·nK,σ+FK,σ^′ nDσ,ǫ·nK,σ^′

.

- Second case – The vectorei is tangent toǫ, andǫis the union of the halves of two primal facesτ and τ^′ such thatσ=K|Lwithτ ∈E(K) andτ^′ ∈E(L). Then:

Fσ,ǫ= 1

2 (FK,τ +FL,τ^′).

Remark 2.1. In both cases, for ǫ = σ|σ^′, the mass flux Fσ,ǫ may be written as Fσ,ǫ = |ǫ|ρǫu˜ǫ, with ρǫ= (ρσ+ρσ^′)andu˜ǫ= (ρσuσ+ρσ^′uσ^′)/(ρσ+ρσ^′) in the first case, andρǫ= (|τ|ρτ+|τ^′|ρτ^′)/(|τ|+|τ^′|) andu˜ǫ= (|τ|ρτuτ+|τ^′|ρτ^′uτ^′)/(|τ|ρτ+|τ^′|ρτ^′)in the second case.

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With this expression of the flux, we may define a discrete divergence operator on the dual cells:

divE(i)(ρ,v) : L^M×H^E,0−→L^E

(ρ,v)7−→divE(i)(ρ,v) = X

σ∈E⁽ⁱ⁾

int

divDσ(ρv)1Dσ, with divDσ(ρ,v) = 1

|Dσ| X

ǫ∈eE(Dσ)

Fσ,ǫ, ∀σ∈E⁽ⁱ⁾

int.

For the definition of the time-derivativeð_t(ρu), an approximation of the density on the dual cellρDσ is defined as:

|Dσ|ρDσ =|DK,σ|ρK+|DL,σ|ρL, σ∈E_int, σ=K|L.

With the above definitions, if (ρ,u)∈L^M×X^E,δtsatisfies the mass balance equation (4b), then the following mass balance on the dual cells holds:

(6) 1

δt(ρⁿ⁺¹_Dσ −ρⁿ_Dσ) + divDσ(ρⁿ⁺¹uⁿ) = 0.

Note that a discrete duality property also holds, in the sense that, for 1≤i≤d, (7) ∀ρ∈L^M,∀v∈HE,0,∀w∈HE(i),0,

Z

Ω

divE(i)(ρ,v)w dx= Z

Ω

ρv· ∇^E(i)wdx, where (ρv)E(i) and∇^E(i)ware vector valued functions of components:

[(ρv)E⁽ⁱ⁾]j= X

ǫ∈Ee(i,j)

ρǫ˜vǫ1Dǫ, [(∇w)E⁽ⁱ⁾]j = X

ǫ∈eE^(i,j),σ=−−→

σ|σ^′

uσ^′−uσ

dǫ 1Dǫ,

withρǫ and ˜vǫ defined in Remark 2.1 andEe^(i,j)={ǫ∈eE⁽ⁱ⁾;ǫ⊥e^(j)}. We finally define thei-th component CE⁽ⁱ⁾(ρu) of the non linear convection operator by:

CE⁽ⁱ⁾(ρ,u) : HE(i),0−→HE(i),0

v7−→CE⁽ⁱ⁾(ρ,u)v= X

σ∈eE⁽ⁱ⁾

int

1

|Dσ| X

ǫ∈E˜(Dσ) ǫ=σ|σ^′

Fσ,ǫ

vσ+vσ^′

2 1Dσ.

and the full (i.e. for all the velocity components) discrete convection operatorC^E(ρ,u), H^E,0−→H^E,0 by CE(ρ,u)v = (CE⁽¹⁾(ρ,u)v1, . . . , CE^(d)(ρu)vd)^t. Let EE be the subspace of HE,0 of divergence-free functions (with respect to the discrete divergence operator). By H¨older’s inequality and [4, Lemma 3.9], there exists CηM >0 (depending only onη^M) such that,∀(ρ,u,v,w)∈L^M×E^E×HE²,0,

|C^E(ρu)v·w| ≤CηMkρkL^∞(Ω)kukL⁴(Ω)^dkvkL⁴(Ω)^dkwk1,E,0

and |C^E(ρu)v·w| ≤CηMkρkL^∞(Ω)kuk1,E,0kvk1,E,0kwk1,E,0. 3. Estimates and convergence analysis

Since the velocity is divergence-free, the mass equation is a transport equation on ρ, so that, thanks to the upwind choice, the following estimate holds:

(8) ρmin≤ρⁿ⁺¹≤ρmax,

and theL²-norm ofρⁿ⁺¹ is lower than the L²-norm of the initial dataρ0, for 0≤n≤N −1. In addition, thanks to (6), any solution to the scheme (4) satisfies the following discrete kinetic energy balance, for 1≤i≤d,σ∈E⁽ⁱ⁾, 0≤n≤N−1,

(9) 1 2δt

ρⁿ⁺¹_Dσ (uⁿ⁺¹_σ )²−ρⁿ_Dσ(uⁿ_σ)²

+ 1

2|Dσ| X

ǫ∈E˜(Dσ) ǫ=σ|σ^′

Fσ,ǫ(ρⁿ⁺¹, uⁿ)uⁿ⁺¹_σ uⁿ⁺¹_σ^′

−(∆u)ⁿ⁺¹_σ uⁿ⁺¹_σ + (ðp)ⁿ⁺¹_σ uⁿ⁺¹_σ −f_σⁿ⁺¹uⁿ⁺¹_σ =− 1

2δtρⁿ_Dσ uⁿ⁺¹_σ −uⁿ_σ2

.

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From this inequality, we obtain estimates on the velocity. For u∈X^E,δt satisfying (4), there existsC >0 depending onu0, ρ0 andf such that,

(10) kukL²(H^E,0)=

NX−1

n=0

δtkuⁿ⁺¹k²1,E,0≤C andkukL^∞(L²)= max

0≤n≤N−1kuⁿ⁺¹kL²(Ω)^d≤C.

These estimates yields the existence of a unique solution to the scheme: indeed, the first equation may be solved separately for ρⁿ⁺¹ and is linear with repect to this unknown and, once ρⁿ⁺¹ is known, the last two equations are a linear generalized Oseen problem for uⁿ⁺¹ and pⁿ⁺¹, which is uniquely solvable thanks to theinf-supstability of the MAC discretization. The convergence of the scheme requires some time compactness. Contrary to the constant density case [4], there is no uniform estimate on the time derivative, and compactness is obtained thanks to the following lemma together with the Fr´echet-Kolmogorov theorem.

Lemma 3.1 (Estimate on the time translates of the velocity). Let u∈X^E,δt andρ∈Y^M,δt and letτ >0 then

(11)

Z T−τ 0

Z

Ω|u(x, t+τ)−u(x, t)|² dx dt≤CηM,T

ρmax

ρmin

(kuk³L²(H^E,0)+ 1)√ τ+δt whereCηM,T >0 only depends onΩ,T,f and on the regularity of the mesh η^M.

Proof. In the continuous case, see e.g. [2, pages 444-452], the estimate (11) is obtained by bounding the termRT−τ

0

R

Ω(ρ(x, t)u(x, t+τ)−ρ(x, t)u(x, t))·w(x, t) dtwithw(x, t) =u(x, t+τ)−u(x, t). However, in the context of the MAC scheme, the components of u are piecewise constant on different meshes so we need to treat the space indices separately. For a giveni= 1, . . . , d, we denote byuandwthei-th component ofu andw, and byρethe piecewise constant function defined by ρ(x, t) =e ρⁿ⁺¹_Dσ for (x, t)∈Dσ×[tntn+1).

We then wish to bound the terms A⁽ⁱ⁾=

Z T−τ 0

(A⁽ⁱ⁾₁ (t) +A⁽ⁱ⁾₂ (t)) dt, with A⁽ⁱ⁾₁ (t) =

Z

Ω

(ρ(x, te +τ)u(x, t+τ)−ρ(x, t)u(x, t))e w(x, t) dx, A⁽ⁱ⁾₂ (t) =

Z

Ω

(ρ(x, t)e −ρ(x, te +τ))u(x, t+τ)w(x, t) dx.

For lack of space, we only deal here with the termA⁽ⁱ⁾₂ (t). Thanks to the mass balance on the dual cells (6) and to the discrete duality formula (7) we have:

A⁽ⁱ⁾₂ (t) =

NX−1

n=1

δt1(t,t+τ)(tn) Z

Ω

divE(i)(˜ρⁿ⁺¹uⁿ)u(·, t+τ)w(·, t) dx

=

NX−1

n=1

δt1(t,t+τ)(tn) Z

Ω

(ρⁿ⁺¹uⁿ)E(i)∇^E(i)(u(·, t+τ)w(·, t)) dx.

Using H¨older’s inequalities and the fact that

NX−1

n=1

δt1(t,t+τ)(tn)≤τ+δt,

A⁽ⁱ⁾₂ (t)≤ρmaxδt^NX⁻¹

n=1

kuⁿkL⁶

¹2 ^NX⁻¹

n=1

1(t,t+τ)(tn)¹2

k∇^E(i)(u(t+τ)w(t))k_L⁶₅

≤ |Ω|¹⁶ρmaxkuk_L²¹²_(L⁶₎(δt+τ)¹²k∇^E(i)(u(t+τ)w(t))k_L³₂. Now, by H¨older’s inequality,

k∇^E(i)(u(·, t+τ)w(·, t))k_L³₂ ≤ k(∇^E(i)u(·, t+τ))w(·, t)k_L³₂ +ku(·, t+τ)∇^E(i)(w(·, t))k_L³₂

≤ k∇^E(i)u(·, t+τ)k²L²+kw(·, t)k²L⁶+k∇^E(i)w(·, t)k²L²+ku(·, t+τ)k²L⁶.

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Therefore, integrating over (0, T −τ) yields that Z T−τ

0

A⁽ⁱ⁾₂ (t) dt≤ |Ω|¹⁶ρmax[τ+δ]¹²kukL²(L⁶)[kukL²(L⁶)+kwkL²(H_E(i),0)+kwkL²(L⁶)+kukL²(H_E(i),0)].

Similar computations for the termRT−τ

0 A⁽ⁱ⁾₁ (t) dtyield the result.

Theorem 3.2 (Convergence of the scheme). Let (δtm)m∈Nand (M_m)m∈N be a sequence of time steps and MAC grids such that δtm → 0 and h^M_m → 0 as m → +∞ ; assume that there exists η > 0 such that η^M_m ≤η for anym∈N. Let(ρm,um) be a solution to (4) forδt=δtmandM=M_m. Then there existsρ¯ withρmin≤ρ¯≤ρmax andu¯ ∈L²(0, T;E(Ω))such that, up to a subsequence:

- the sequence (um)m∈Nconverges to u¯ inL²(0, T;L²(Ω)^d), - the sequence (ρm)m∈N converges to ρ¯in∈L²(0, T;L²(Ω)), - (¯ρ,u)¯ is a solution to the weak formulation (2) and (3).

Sketch of proof:

– Thanks to (8), there exists a subsequence of (ρm)m∈N star-weakly converging to some ¯ρin L^∞(Ω× (0, T)); thanks to (10) and (11), there exists a subsequence of (um)m∈N converging to some ¯u in L²(0, T; (L²(Ω)^d).

– Passing to the limit in (4b) yields that (¯ρ,¯u) satisfies (2).

– The strong convergence of the approximate densities is then obtained thanks to theL²estimates for ρin both the discrete and continuous case [7, Proposition 8.7].

– Passing to the limit in (4c) yields that (¯ρ,u) satisfies (3).¯

– We finally obtain that ¯u∈L²(0, T;E(Ω)), whereE(Ω) ={v∈H₀¹(Ω) s.t. divv = 0}, as in [4, Proof of Theorem 4.3].

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Math. Stat., pages 627–635. Springer, Cham, 2014.

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Int. J. Finite Vol., 7(2):6, 2010.

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Mathematics of Computation, accepted for publication, 2016.

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Anal., 21(5):1093–1117, 1990.

I2M UMR 7373, Aix-Marseille Universit´e, CNRS, ´Ecole Centrale de Marseille.

E-mail address:[email protected]

I2M UMR 7373, Aix-Marseille Universit´e, CNRS, ´Ecole Centrale de Marseille.

E-mail address:[email protected]

Institut de Radioprotection et de Sˆuret´e Nucl´eaire (IRSN), Saint-Paul-lez-Durance, 13115, France.

E-mail address:[email protected] University of Skikda, Algeria.

E-mail address:[email protected]

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